On saturation of the Cramér Rao Bound for Sparse Bayesian Learning

This paper analyzes the Cramér-Rao Bound associated with the estimation of certain sparse hyper-parameters in the Sparse Bayesian Learning (SBL) framework, that crucially control the sparsity of the desired signal. The CRB is shown to exhibit saturation with respect to the number of measurements, i.e., it can be lower bounded by a non-negative quantity that does not go to zero even when the number of measurements tends to infinity. Moreover, the CRB corresponding to the nonzero and zero elements of the sparse hyper-parameter can exhibit different behaviors. While the CRB for the non-zero elements always saturate regardless of the type of dictionary, saturation of the CRB for zero elements provably happens when the dictionary has normalized columns. For an unnormalized dictionary, singular values of certain sub-dictionaries determine if saturation can happen, prompting future research into this interesting phenomenon.